Chapter 12: Descriptive Statistics

1. Chapter 12: Descriptive Statistics • Objectives • Describe the process of tabulating and coding data. • Define frequency and central tendency, and differentiate among mean, median, and mode. • Define variability, and differentiate among the range, quartile deviation, variance, and standard deviation.

2. Chapter 12: Descriptive Statistics • Describe the major characteristics of normal and skewed distributions. • Define and differentiate among measures of relative position, including percentile ranks and standard scores. • Define and differentiate among two measures of relationship, the Pearson r and the Spearman rho.

3. Descriptive Statistics • Statistics is a set of procedures for describing, synthesizing, analyzing, and interpreting quantitative data. • The mean is an example of a statistic. • One can calculate statistics by hand or can use the assistance of statistical programs. • Excel, SPSS, and many other programs exist. Some programs are also available on the Web to analyze datasets.

4. Preparing Data for Analysis • After data are collected, the first step toward analysis involves converting behavioral responses into a numerical system or categorical organization. • It is critical that all data are scored accurately and consistently. • Data scoring should be double-checked for consistency and accuracy (i.e., at least 25% of all cases should be checked).

5. Preparing Data for Analysis • Open-ended items should be scored by two scorers to check reliability. • All data scoring and coding procedures should be documented and reported in the written report.

6. Preparing Data for Analysis • After instruments are scored, the resulting data are tabulated and entered into a spreadsheet. • Tabulation involves organizing the data systematically (e.g., by participant).

7. Preparing Data for Analysis • In this potential dataset, • ID represents participant number • Cond. is the experimental condition (1 or 2) • Gender is represented by female=1; male=2 • Achievement (Ach) and motivation (Mot) were also variables assessed

8. Types of Descriptive Statistics • After data are tabulated and entered, the next step is to conduct descriptive statistics to summarize data. • In some studies, only descriptive statistics will be conducted. • If the indices are calculated for a sample, they are referred to as statistics. • If indices are calculated for the entire population, they are referred to as parameters.

9. Types of Descriptive Statistics • Frequencies • The frequency refers to the number of times something occurs. • Frequencies are often used to describe categorical data. • We might want to have frequency counts of how many males and females were in a study or how many participants were in each condition. • Frequency counts are not as helpful in describing interval and ratio data.

10. Measures of Central Tendency • Measures of central tendency are indices that represent a typical score among a group of scores. • Measures of central tendency provide a way to describe a dataset with a single number.

11. Measures of Central Tendency • The three most common measures of central tendency are the mean, median, and mode. • Mean: Appropriate for describing interval or ratio data • Median: Appropriate for describing ordinal data • Mode: Appropriate for describing nominal data

12. Measures of Central Tendency • The mean is the most commonly used measure of central tendency. • The formula for the mean is: X= ∑Xi/n • To calculate the mean, all the scores are summed and then divided by the number of scores.

13. The Mean or Average Value Example What is the mean of 4 3 6 8 4 Mean = (4+3+6+8+4)/5 = 5

14. The Median Value • The median is the midpoint in a distribution: 50% of the scores are above the median and 50% are below the median. • To determine the median, all scores are listed in order of value. • If the total number of scores is odd, the median is the middle score.

15. The Median Value • If the total number of scores is even, the median is halfway between the two middle scores. • Median values are useful when there is large variance in a distribution.

16. The Median Value Example 1: Use the previous data put in order 3 4 4 6 8 The Median is (the middle value) = 4 Example 2: 3 4 4 6 8 9 The Median is (4+6/2) = 5

17. The Mode • The mode is the most frequently occurring score in a distribution. • The mode is established by looking at a set of scores or at a graph of scores and determining which score occurs most frequently. • The mode is of limited value. • Some distributions have more than one mode (e.g., bi-modal, or multi-modal distributions)

18. Measures of Central Tendency • Deciding among measures of central tendency • Generally the mean is most preferred. • The mean takes all scores into account. • The mean, however, is greatly influenced by extreme scores- outlying data. • When there are extreme scores present in a distribution, the median is a better measure of central tendency.

19. Measures of Variability • Measures of variability provide an index of the degree of spread in a distribution of scores. • Measures of variability are critical to examine and report because some distributions may be very different but yet still have the same mean or median.

20. Measures of Variability • Three common measures of variability are the range, quartile deviation, and standard deviation. • Range: The difference between the highest and lowest score. • The range is not a stable measure. • The range is quickly determined.

21. Measures of Variability • Quartile Deviation: One half the difference between the upper quartile and the lower quartile in a distribution. • By subtracting the cutoff point for the lower quartile from the cutoff point for the upper quartile and then dividing by two we obtain a measure of variability. • A small number indicates little variability and illustrates that the scores are close together.

22. Measures of Variability • Variance: The amount of spread among scores. If the variance is small the scores are close together. If the variance is large the scores are spread out. • Calculation of the variance shows how far each score is from the mean. • The formula for the variance is: ∑(X–X)2/n

23. Measures of Variability • Standard deviation: The square root of the variance. • The standard deviation is used with interval and ratio data. • The standard deviation is the most commonly used measure of variability.

24. Measures of Variability • If the mean and the standard deviation are known, the distribution can be described fairly well. • SD represents the standard deviation of a sample and the symbol  (i.e., the Greek lower case sigma) represents the standard deviation of the population.

25. Standard Deviation – 3 cases with the same Mean